Eaman Eftekhary
We show that among Seifert fibered integer homology spheres, Poincare sphere (with either orientation) is the only non-trivial example which has trivial Heegaard Floer homology. Together with an earlier result, this shows that if an integer homology sphere has trivial Heegaard Floer homology, then it is a connected sum of a number of Poincare spheres and hyperbolic homology spheres.
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Seifert fibered surgeries which do not arise from primitive/Seifert-fibered constructions
我们构建了两个无限的家庭结，每个家庭都承认一个塞弗特纤维手术，这些手术都没有来自迪恩的原始/塞弗特纤维建设。这否定了一个猜想，即所有的塞弗特纤维手术都来自迪恩的原始/塞弗特纤维结构。（-3，3，5）-椒盐卷饼结属于这两个无限族。
A Heegaard splitting of a $3$-manifold is flippable if there is an isotopy that interchanges the two sides of the Heegaard splitting. We explore which Heegaard splittings of Seifert fibered spaces are flippable.
We find an infinite family of Seifert fibered surgeries on strongly invertible knots which do not have primitive/Seifert positions. Each member of the family is obtained from a trefoil knot after alternate twists along a pair of seiferters for a Seifert fibered surgery on a trefoil knot.
We define the surface complex for $3$-manifolds and embark on a case study in the arena of Seifert fibered spaces. The base orbifold of a Seifert fibered space captures some of the topology of the Seifert fibered space, so, not surprisingly, the surface complex of a Seifert fibered space always contains a subcomplex isomorphic to the curve complex of the base orbifold.
This paper concerns the problem of existence of taut foliations among 3-manifolds. Since the contribution of David Gabai, we know that closed 3-manifolds with non-trivial second homology group admit a taut foliations. The essential part of this paper focuses on Seifert fibered homology 3-spheres. The result is quite different if they are integral or rational but non-integral homology 3-spheres. Concerning integral homology 3-spheres, we prove that all but the 3-sphere and the Poincaré 3-sphere admit a taut foliation. Concerning non-integral homology 3-spheres, we prove there are infinitely man…
Heegaard splittings of Seifert manifolds without an orientable base space
利用Heath，Moriah和Schultens的结果证明了具有不可定向底空间的可定向Seifert流形的不可约Heegaard分裂是垂直的或水平的.
Horizontal Heegaard splittings of Seifert fibered spaces
本文证明了，如果具有可定向亏格$g$底空间的可定向Seifert纤维空间$M$允许强不可约水平Heegaard分裂，则强不可约水平Heegaard分裂的合痕类与$\mathbf{Z}^{2g}$的元素之间存在一一对应.该对应关系由每个Heegaard分裂与$M$中的$2g$不可压缩环面的集合的交点的斜率确定。我们还表明，有塞弗特纤维空间与无限多的非同位素Heegaard分裂，确定尼尔森等价生成系统的基本群$M$。
We call a pair (K, m) of a knot K in the 3-sphere S^3 and an integer m a Seifert fibered surgery if m-surgery on K yields a Seifert fiber space. For most known Seifert fibered surgeries (K, m), K can be embedded in a genus 2 Heegaard surface of S^3 in a primitive/Seifert position, the concept introduced by Dean as a natural extension of primitive/primitive position defined by Berge. Recently Guntel has given an infinite family of Seifert fibered surgeries each of which has distinct primitive/Seifert positions. In this paper we give yet other infinite families of Seifert fibered surgeries with …
Heegaard gradient of Seifert fibered 3-manifold
紧三维流形的下确界Heegaard梯度是由Marc Lackenby定义和研究的，其方法是为了解决著名的虚Haken猜想。作为有指导意义的例子，我们考虑塞弗特纤维3-流形，并表明，塞弗特纤维3-流形有零的下底Heegaard梯度当且仅当它几乎纤维的圆或表面以外的2-球。对于Seifert纤维3-流形的有限覆盖集，给出了其下确界Heegaard梯度为零的一个充要条件.
We establish three rank inequalities for the reduced flavor of Heegaard Floer homology of Seifert fibered integral homology spheres. Combining these inequalities with the known classifications of non-zero degree maps between Seifert fibered spaces, we prove that a map f from Y' to Y between Seifert homology spheres yields the inequality that |deg f| rank HFred(Y) is at most rank HFred(Y'). These inequalities are also applied in conjunction with an algorithm of Nemethi to give a method to solve the botany problem for the Heegaard Floer homology of these manifolds.
We show that if a Montesinos knot admits a Dehn surgery yielding a toroidal Seifert fibered 3-manifold, then the knot is the trefoil knot and the surgery slope is 0.
Using the mapping cone of a rational surgery, we give several obstructions for Seifert fibered surgeries, including obstructions on the Alexander polynomial, the knot Floer homology, the surgery coefficient and the Seifert and four-ball genus of the knot.
Exceptional Dehn surgeries on arborescent knots have been classified except for Seifert fibered surgeries on Montesinos knots of length 3. There are infinitely many of them as it is known that 4n+6 and 4n+7 surgeries on a (-2, 3, 2n+1) pretzel knot are Seifert fibered. It will be shown that there are only finitely many others. A list of 20 surgeries will be given and proved to be Seifert fibered. We conjecture that this is a complete list.
We give a new criterion for a given knot to be a Montesinos knot by using the Rasmussen invariant and the signature. We apply the criterion to study Seifert fibered surgery on a strongly invertible knot, and show that a $(p,q,q)$-pretzel knot with integers $p,q \ge 2$ admits no Seifert fibered surgery.
We will use immersed surfaces to study Seifert fibered surgery on Montesinos knots, and show that if $\frac 1{q_1-1} + \frac 1{q_2-1} + \frac 1{q_3-1} \leq 1$ then a Montesinos knot $K(\frac{p_1}{q_1}, \frac{p_2}{q_2}, \frac{p_3}{q_3})$ admits no atoroidal Seifert fibered surgery.
The twisted torus knots lie on the standard genus 2 Heegaard surface for $S^3$, as do the primitive/primitive and primitive/Seifert knots. It is known that primitive/primitive knots are fibered, and that not all primitive/Seifert knots are fibered. Since there is a wealth of primitive/Seifert knots that are twisted torus knots, we consider the twisted torus knots to partially answer the question of which primitive/Seifert knots are fibered. A braid computation shows that a particular family of twisted torus knots is fibered, and that computation is then used to generalize the results of a prev…
We study compatible contact structures of fibered Seifert multilinks in homology 3-spheres and especially give a necessary and sufficient condition for the contact structure to be tight in the case where the Seifert fibration is positively twisted. As a corollary we determine the strongly quasipositivity of fibered Seifert links in the 3-sphere. We also study the compatible contact structures of cablings along links in any 3-manifolds.
Non-isotopic Heegaard splittings of Seifert fibered spaces
我们找到了环面三维流形的强不可约Heegaard分裂的合痕类的一个几何不变量。结合这一不变量与R魏德曼定理，证明在这里的附录中，我们表明，一个封闭的，完全定向塞弗特纤维空间M有无穷多个合痕类的Heegaard分裂的相同属当且仅当M有一个不可约的，水平Heegaard分裂，有一个基地orbifold的正亏格，是不是一个圆丛。这正好刻画了哪些塞弗特纤维空间满足瓦尔德豪森猜想的逆。
Suppose $α$ and $R$ are disjoint simple closed curves in the boundary of a genus two handlebody $H$ such that $H[R]$ embeds in $S^3$ as the exterior of a hyperbolic knot $k$(thus, $k$ is a tunnel-number-one knot), and $α$ is Seifert in $H$(i.e., a 2-handle addition $H[α]$ is a Seifert-fibered space) and not the meridian of $H[R]$. Then for a slope $γ$ of $k$ represented by $α$, $γ$-Dehn surgery $k(γ)$ is a Seifert-fibered space. Such a construction of Seifert-fibered Dehn surgeries generalizes that of Seifert-fibered Dehn surgeries arising from primtive/Seifert positions of a knot, which was i…
In this paper, we define the primitive/Seifert-fibered property for a knot in S^3. If satisfied, the property ensures that the knot has a Dehn surgery that yields a small Seifert-fibered space (i.e. base S^2 and three or fewer critical fibers). Next we describe the twisted torus knots, which provide an abundance of examples of primitive/Seifert-fibered knots. By analyzing the twisted torus knots, we prove that nearly all possible triples of multiplicities of the critical fibers arise via Dehn surgery on primitive/Seifert-fibered knots.
We give a complete classification of toroidal Seifert fibered surgeries on alternating knots. Precisely, we show that if an alternating knot admits a toroidal Seifert fibered surgery, then the knot is either the trefoil knot and the surgery slope is zero, or the connected sum of a (2,p)-torus knot and a (2,q)-torus knot and the surgery slope is 2(p+q) with |p|, |q| at least three.
We derive a formula for the $\barμ$-invariant of a Seifert fibered homology sphere in terms of the eta-invariant of its Dirac operator. As a consequence, we obtain a vanishing result for the index of certain Dirac operators on plumbed 4-manifolds bounding such spheres.
Explicit horizontal open books on some Seifert fibered 3--manifolds
我们描述了一些塞弗特纤维3-流形上的显式水平开书。我们表明，切触结构兼容这些水平打开的书籍是斯坦填充和水平以及。此外，我们还绘制了这些切触结构的手术图。
Seifert Fibered Spaces: Notes for a course given in the Spring of 1993
一个学期课程的笔记。注记包括紧致三维Seifert纤维空间的描述和具有非空边界的紧致三维Seifert纤维空间到同胚的分类。
We examine surgery on a knot in $S^3$ to determine surgery obstructions to Seifert fibered integral homology spheres. We find such surgery obstructions using Heegaard Floer, Knot Floer homology and the mapping cone formula for computing Heegaard Floer homology of surgery on a knot. Here however, we take a different approach and use the number of singular fibers of a Seifert fibered integral homology sphere, which is the toroidal structure, to find obstructions. This approach allows us to show that genus one knots cannot yield Seifert fibered integral homology spheres with six or more singular …
One measure of the complexity of a 3-manifold is its triangulation complexity: the minimal number of tetrahedra in a triangulation of it. A natural question is whether we can relate this quantity to its topology. We determine the triangulation complexity of Seifert fibered spaces with non-empty boundary in terms of their Seifert data, up to a multiplicative constant. We also show that all singular fibres of such a Seifert fibered space (aside from those of multiplicity two) can be made simplicial in the 79th barycentric subdivision of any triangulation of it.
We prove that if a fibered knot $K$ with genus greater than one in a three-manifold $M$ has a sufficiently complicated monodromy, then $K$ induces a minimal genus Heegaard splitting $P$ that is unique up to isotopy, and small genus Heegaard splittings of $M$ are stabilizations of $P$. We provide a complexity bound in terms of the Heegaard genus of $M$. We also provide global complexity bounds for fibered knots in the three-sphere and lens spaces.
In this paper, we add examples to Goeritz groups, the mapping class groups of given Heegaard splittings of 3-manifolds. We focus on a Heegaard splitting of genus two of a Seifert manifold whose base orbifold is sphere with three exceptional points of sufficiently complex coefficients, where "sufficiently complex" means that every surgery coefficient p_{l} over q_{l} of each exceptional fiber (in a surgery description) satisfies that q_{l} is congruent to neither +1 or -1 modulo p_{l}.
We study spanning sets for the Kauffman bracket skein module $\mathcal{S}(M,\mathbb{Q}(A))$ of orientable Seifert fibered spaces with orientable base and non-empty boundary. As a consequence, we show that the KBSM of such manifolds is a finitely generated $\mathcal{S}(\partial M, \mathbb{Q}(A))$-module.
We introduce generalized arrow diagrams and generalized Reidemeister moves for diagrams of links in Seifert fibered spaces. We give a presentation of the fundamental group of the link complement. As a corollary we are able to compute the first homology group of the complement and the twisted Alexander polynomials of the link.
Rationally null-homologous links in Seifert fibered spaces may be represented combinatorially via labeled diagrams. We introduce an additional condition on a labeled link diagram and prove that it is equivalent to the existence of a rational Seifert surface for the link. In the case when this condition is satisfied, we generalize Seifert's algorithm to explicitly construct a rational Seifert surface for any rationally null-homologous knot. As an application of the techniques developed in the paper, we derive closed formulae for the rational Thurston-Bennequin and rotation numbers of a Legendri…
We establish an inequality which gives strong restrictions on when the standard definite plumbing intersection lattice of a Seifert fibered space over $S^2$ can embed into a standard diagonal lattice, and give two applications. First, we answer a question of Neumann-Zagier on the relationship between Donaldson's theorem and Fintushel-Stern's $R$-invariant. We also give a short proof of the characterisation of Seifert fibered spaces which smoothly bound rational homology $S^1 \times D^3$'s.
Symplectic rational blow-down along Seifert fibered 3-manifolds
我们证明了第二作者Szabo和Wahl所发现的沿某些Seifert纤维3-流形的有理blow-down格式实际上是辛运算。
Classification of tight contact structures on some Seifert fibered manifolds: I
本文通过 Legendrian 手术构造接触结构获得下界，并利用凸曲面理论获得上界，对具有四条例外纤维的某些 Seifert 纤维化流形上零 Giroux 挠率的紧接触结构进行了分类。
It is known that every closed oriented 3-manifold is homology cobordant to a hyperbolic 3-manifold. By contrast we show that many homology cobordism classes contain no Seifert fibered 3-manifold. This is accomplished by determining the isomorphism type of the rational cohomology ring of all Seifert fibered 3-manifolds with no 2-torsion in their first homology. Then we exhibit families of examples of 3-manifolds (obtained by surgery on links), with fixed linking form and cohomology ring, that are not homology cobordant to any Seifert fibered space (as shown by their rational cohomology ring). T…
The Ozsvath-Szabo contact invariant is a complete classification invariant for tight contact structures on small Seifert fibered 3-manifolds which are L-spaces.
We give a surgery formula for the asymptotic behavior of the sequence given by the logarithm of the higher dimensional Reidemeister torsion. Applying the resulting formula to Seifert fibered spaces, we show that the growth of the sequences has the same order as the indices and we give the explicit values for the limits of the leading coefficients. There are finitely many possibilities as the limits of the leading coefficients for a Seifert fibered space. We also show that the maximum is given by the product of (-log 2) and the Euler characteristic of the base orbifold for a Seifert fibered spa…
We study small Seifert possibly chiral cosmetic surgeries on not necessarily null-homologous knot in rational homology spheres. Using $PSL_2(\mathbb{C})$-character variety theory we give a sharp bound on the number of slopes producing the same small Seifert manifold if the ambient manifold satisfies some representation theoretic conditions.
The main result of this paper asserts that if a Seifert fibered 4-manifold has nonzero Seiberg-Witten invariant, the homotopy class of regular fibers has infinite order. This is a nontrivial obstruction to smooth circle actions; as applications, we show how to destroy smooth circle actions on a 4-manifold by knot surgery, without changing the integral homology, intersection form, and even the Seiberg-Witten invariant. Results concerning classification of Seifert fibered complex surfaces or symplectic 4-manifolds are included. We also show that every smooth circle action on the 4-torus is smoot…
Let $\barγ$ be a link in a Seifert fibered space $M$ over a hyperbolic $2$-orbifolds $\mathcal O$ that projects injectively to a filling multicurve of closed geodesics $γ$ in $\mathcal O.$ We prove that the complement $M_{\barγ}$ of $\barγ$ in $M$ admits a hyperbolic structure of finite volume and give combinatorial bounds of its volume.
We compute the connected Heegaard Floer homology (defined by Hendricks, Hom, and Lidman) for a large class of 3-manifolds, including all linear combinations of Seifert fibered homology spheres. We show that for such manifolds, the connected Floer homology completely determines the local equivalence class of the associated $ι$-complex. Some identities relating the rank of the connected Floer homology to the Rokhlin invariant and the Neumann-Siebenmann invariant are also derived. Our computations are based on combinatorial models inspired by the work of Némethi on lattice homology.
We complete the classification of hyperbolic pretzel knots admitting Seifert fibered surgeries. This is the final step in understanding all exceptional surgeries on hyperbolic pretzel knots. We also present results toward similar classifications for non-pretzel Montesinos knots of length three.
On the existence of tight contact structures on Seifert fibered 3-manifolds
我们确定了具有正紧切触结构的闭定向Seifert纤维三维流形。我们的主要工具是一个新的接触Ozsvath-Szabo不变量的非零准则。
All integral slopes can be Seifert fibered slopes for hyperbolic knots
对于三维球面S^3中的双曲纽结，哪些斜率可以或不可以表现为塞弗特纤维斜率？证明了如果对S^3中的双曲纽结进行r-外科手术得到一个塞弗特纤维空间，则r是一个整数。我们证明了对于每个整数n，在S^3中存在一个隧道数为1的双曲纽结K_n，使得对K_n的n次手术产生一个小的塞弗特纤维空间。
We give a necessary and sufficient condition for a simple closed curve on the boundary of a genus two handlebody to decompose the handlebody into (torus with one boundary component times [0,1]. We use this condition to decide whether a simple closed curve on a genus two Heegaard surface is a GOF-knot (genus one fibered knot) which induces the Heegaard splitting. By using this, we determine the number and the positions with respect to the Heegaard splittings of GOF-knots in the 3-manifolds with reducible genus two Heegaard splittings. This is another proof of results of Morimoto and Baker.
Seifert fibered 3-manifolds and Turaev-Viro invariants volume conjecture
研究在根q=e^{2πi/r}处定向Seifert纤维化3-流形的Turaev-Viro不变量的大r渐近行为，证明了一大类带边与无边定向Seifert纤维化3-流形的体积猜想。
We characterize the closed, oriented, Seifert fibered 3-manifolds which are oriented boundaries of Stein manifolds. We also show that for this class of 3-manifolds the existence of Stein fillings is equivalent to the existence of symplectic fillings.
Seiberg--Witten Monopoles on Seifert Fibered Spaces
本文研究了Seifert纤维空间的Seiberg-Witten规范理论。在这些三个流形的单极，为一个特定的选择度量和扰动，完全描述。单极子之间的梯度流线识别与相关的直纹表面上的全纯数据，并计算出这种流动的尺寸公式。
Seifert fibered contact three-manifolds via surgery
使用接触外科手术，我们定义了家庭的切触结构某些塞弗特纤维三流形。我们证明了所有这些切触结构是紧的接触Ozsath-Szabo不变量。我们用这些例子表明，给定一个自然数n，存在一个塞弗特纤维三流形携带至少n个两两非同构的紧，不填充的切触结构。
We define a differential graded algebra associated to Legendrian knots in Seifert fibered spaces with transverse contact structures. This construction is distinguished from other combinatorial realizations of contact homology invariants by the existence of orbifold points in the Reeb orbit space of the contact manifold. These orbifold points are images of the exceptional fibers of the Seifert fibered manifold, and they play a key role in the definitions of the differential and the grading, as well as the proof of invariance. We apply the invariant to distinguish Legendrian knots whose homology…
Using an obstruction based on Donaldson's theorem, we derive strong restrictions on when a Seifert fibered space $Y = F(e; \frac{p_1}{q_1}, \ldots, \frac{p_k}{q_k})$ over an orientable base surface $F$ can smoothly embed in $S^4$. This allows us to classify precisely when $Y$ smoothly embeds provided $e > k/2$, where $e$ is the normalized central weight and $k$ is the number of singular fibers. Based on these results and an analysis of the Neumann-Siebenmann invariant $\overlineμ$, we make some conjectures concerning Seifert fibered spaces which embed in $S^4$. Finally, we also provide some ap…
On Heegaard Floer homology and Seifert fibered surgeries
我们探讨了某些限制结在三个领域承认非平凡的塞弗特纤维手术。这些限制源于塞弗特纤维空间的Heegaard弗洛尔同调，因此它们对这种纽结的Alexander多项式和纽结的弗洛尔同调都有影响。特别是，我们表明，某些多项式从来没有亚历山大多项式的结承认同源三球塞弗特纤维手术。另一方面，纽结弗洛尔同调限制也适用于亚历山大多项式没有给出信息的情况，例如木下寺坂纽结。
Comparing Heegaard and JSJ structures of orientable 3-manifolds
不可约闭可定向3-流形的Heegaard亏格g限制了流形通过其标准环面的E-Shalen-Johannson分解中出现的块的数量和复杂性。例如，如果p个互补组分不是塞弗特纤维化的，则p < g。这一结果推广了Kobayashi的工作。Heegaard亏格g也对塞弗特块的复杂性给出了明确的界限。例如，如果Seifert片的底空间的并具有欧拉示性数X，且Seifert片中共有f个例外纤维，则f-X不大于3g - 3 -p.
It is well known that, among closed spherical Seifert three-manifolds, only lens spaces and prism manifolds admit several Seifert fibrations which are not equivalent up to diffeomorphism. Moreover the former admit infinitely many fibrations, and the latter exactly two. In this work, we analyse the non-uniqueness phenomenon for orbifold Seifert fibrations. For any closed spherical Seifert three-orbifold, we determine the number of its inequivalent fibrations. When these are in a finite number (in fact, at most three) we provide a complete list. In case of infinitely many fibrations, we describe…
The SL(2,C) Casson invariant for Seifert fibered homology spheres and surgeries on twist knots
利用SL（2，C）特征簇与秩为2的抛物Higgs丛的模空间之间的对应关系，得到了Seifert纤维同调3-球面的SL（2，C）Casson不变量的一个简单的闭公式.这些结果然后被用来推导出的不变量的Dehn手术的扭曲结结合计算的Culler-Shalen范数与SL（2，C）卡森不变量的手术公式。
In this paper we provide the classification of tight contact structures on some small Seifert fibered manifolds. As an application of this classification, combined with work of Lekili in \cite{L2010}, we obtain infinitely many counterexamples to a question of Honda-Kazez-Matić that asks whether a right-veering, non-destabilizable open book necessarily supports a tight contact structure.
Reversible and other generalised torsion elements in Seifert-fibered spaces
本文分类Fuchsian群中的可逆元，进而确定Seifert纤维群及3股辫群中的可逆元，并研究PSL(2,Z)中的广义3-挠元以分析其在Seifert纤维空间和3股辫群中的存在性。
Tight contact structures on some small Seifert fibered 3--manifolds
本文对小Seifert纤维3-流形M（-1; r_1，r_2，r_3）上的紧切触结构进行了分类，其中ri为（0，1），r_1，r_2为\geq 1/2.将凸曲面理论与接触Ozsvath-Szabo不变量的计算相结合得到了这一结果。我们还表明，一些紧切触结构上考虑的流形是不可填充的，证明使用Heegaard Floer理论。
Let M be a closed orientable Seifert fibered 3-manifold with a hyperbolic base 2-orbifold, or equivalently, admitting a geometry modeled on H^2 \times R or the universal cover of SL(2,R). Our main result is that the connected component of the identity map in the diffeomorphism group Diff(M) is either contractible or homotopy equivalent to the circle, according as the center of the fundamental group of M is trivial or infinite cyclic. Apart from the remaining case of non-Haken infranilmanifolds, this completes the homeomorphism classifications of Diff(M) and of the space of Seifert fiberings of…
Toledo invariants of Higgs bundles on elliptic surfaces associated to base orbifolds of Seifert fibered homology 3-spheres
对于从塞弗特纤维同调3-球面的基轨道的轨道基本群到李群U（2，1）的半单表示空间中的每个连通分支，我们关联一个称为“轨道Toledo不变量”的实数。对于每一个这样的轨道，都存在一个椭圆形表面，称为Dolgachev表面。使用这些Dolgachev表面上的希格斯束的理论，我们明确地计算所采取的轨道托莱多不变量的所有值。
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